Internal organs often need to be positioned and/or stabilized for specific surgical medical procedures. In some procedures, the surgeon may wish to move or orient the organ so that surgery may be performed upon it more easily. In other cases, the surgeon may wish to hold the organ, or a portion of it, immobile so that the organ will not move during the surgical procedure so that the area that is being operated on is stable to ensure the accuracy of the surgeon's work. In still other cases, the surgeon may simply want to move the organ out of the way to improve access or visualization. There is a desire in the art to avoid having the use of a tissue positioner cause damage to the tissue that is being grasped. This is particularly important when operating on an internal organ such as the heart. What is needed is a tissue positioner that can be used in both open surgical procedures and through small incisions and ports, that has a small profile thereby increasing the overall visibility of the surgeon, and which may include multiple suction or contact points to limit the gross tissue damage per area by distributing vacuum force over multiple attachments.